How to export texture after editing nodes/uv
I think it would be easier to adjust UV map, then to change distorted texture, but you can do that, by baking Diffuse color to new texture, then exporting it as an image.Like this:

Below you will see, how it needs to be done in Blender Internal in your particular example.As the shirt is made out of 3 different models, I joined them for baking purpose, after that you won't need the duplicate, so you can delete it, cause If the game expects 3 meshes as a shirt input, then probably exporting 1 mesh will not work.